WASHINGTON — President Donald Trump is seeing steels mills spring up where they aren’t and cleaner air than his government is tracking in its records.

Over the past week, his comment that people need photo IDs to shop in stores displayed a misunderstanding of the marketplace obvious to any average shopper with cash or a credit card.

A sampling of comments by Trump and his officials, spanning NATO, the Russia investigation, environmental matters and more:

CLEAN AIR TRUMP, on air quality in the U.S.: “It’s the best it’s ever been.” — Pennsylvania rally Thursday.

THE FACTS: Not true, going by the key measure of air quality. The EPA’s air quality index shows a worsening since 2014, the best year as measured by the number of days with bad air.

For that index, 35 cities reported unhealthy air for a total of 599 days in 2014. That went up to 729 days in 2017, the worst year since 2012 (1,297 days). The index measures ozone and soot. ___

JOBS TRUMP: “U.S. Steel is opening up seven plants.” — remarks Thursday at the Pennsylvania rally. On Tuesday: “Thanks to our tariffs, our steel workers are back on the job, American steel mills are back open for business … U.S. Steel just announced that they’re building six new steel mills.” — Florida rally.

THE FACTS: No, U.S. Steel has not announced six, or seven, new steel mills. A spokeswoman for the Pittsburgh-based company, Meghan Cox, declined to comment on Trump’s claim, only making clear that any “operational changes” such as the opening of new mills would be “publicly announced” and “made available on our website” if it occurred. ___

AUTO SAFETY EPA, citing potential benefits from freezing Obama-era mileage standards: “Increased vehicle affordability leading to increased driving of newer, safer, more efficient and cleaner vehicles. … Over 12,000 fewer crash fatalities over the lifetimes of all vehicles built through model year 2029. Up to 1,000 lives saved annually.” — information sheet released Thursday.

THE FACTS: The claimed safety benefits are unverifiable and probably overstated. While newer vehicles are safer due to better engineering and safety features such as more air bags, automatic emergency braking and blind spot detection, auto safety experts say the difference between vehicles made 10 years ago and now isn’t that big and the number of lives saved can’t really be calculated. Decade-old vehicles have anti-lock brakes and electronic stability control that stop drivers from losing control, two major safety advances. ___

EPA assistant administrator Bill Wehrum: “We’ll leave the standards at a place where we’re not imposing undue costs on manufacturers.” — news briefing Thursday. THE FACTS: Insulating U.S. manufacturers is not easy to do. Even if the U.S. freezes its mileage requirements, the European Union, China, Japan and other nations will continue to increase theirs, which already are more stringent. Because most automakers sell vehicles worldwide, they’ll have to develop new technology such as electric cars anyway to satisfy other markets. The U.S. may not get the new technology as quickly as elsewhere. ___

VOTER ID TRUMP: “We believe that only American citizens should vote in American elections, which is why the time has come for voter ID like everything else. If you go out and you want to buy groceries, you need a picture on a card, you need ID. You go out, you want to buy anything, you need ID, you need your picture.” — remarks Tuesday. THE FACTS: As shoppers know, no photo is required to purchase items at retail stores with cash or to make routine purchases with credit or debit cards.

Identifications are required to purchase limited items such as alcohol, cigarettes or cold medicine and in rapidly declining situations in which a customer opts to pay with a personal check. According to the National Grocers Association’s most recent data, the use of checks as a percentage of total transactions dropped from 33 percent in 2000 to 6 percent in 2015, due in part to the popularity of debit cards, which use PIN codes. The group’s members are independent food retailers, family-owned or privately held, both large and small. ___

WHITE HOUSE PRESS SECRETARY SARAH HUCKABEE SANDERS: “He’s not saying every time he went in; he said when you go to the grocery store.” — press briefing Wednesday. THE FACTS: Actually, Trump did claim, erroneously, that photo IDs are required whenever “you want to buy anything,” not only in limited cases. Asked when Trump last bought groceries, Sanders responded, “I’m not sure. I’m not sure why that matters, either.” ___

NATO TRUMP: “I went to NATO. And NATO was essentially going out of business ’cause people weren’t paying and it was going down, down, down.” On NATO Secretary General Jens Stoltenberg: “He said we couldn’t collect money until President Trump came along. And he said last year we collected $44 billion. And this year the money is pouring in. … So the bottom line is the NATO countries are now paying a lot more money.” — news conference Monday with Italian Prime Minister Giuseppe Conte.

THE FACTS: Countries don’t pay to be in NATO and don’t owe the organization anything other than contributions to a largely administrative fund that Trump is not talking about. Member countries are not in debt to NATO. Money is “not pouring in” now. Collections have not increased, as he asserted. Trump’s actual beef is with how much NATO countries spend on their own military budgets. The Trump administration is not the first to push countries in NATO to spend more on their own armed forces to lessen their dependence on the U.S. In fact, it was in 2014, during the Obama administration, that NATO members agreed to move “toward” spending 2 percent of their gross domestic product on their own defense by 2024.

The somewhat-vague commitment was made as a response to Russia’s actions in Ukraine and its annexation of Crimea. No one expected all allies would immediately move to 2 percent; the increases were to be gradual. ___

VETERANS TRUMP: “We passed the biggest VA reform in half a century, Veteran’s Choice. If our veterans can’t get the care they need from the VA, they will have the right to go see a private doctor.” — remarks Tuesday in Tampa.

THE FACTS: Trump’s suggestion that veterans can get care immediately under the private-sector Veterans Choice program and without restriction is misleading.

Before veterans have the right to see a private doctor, they must meet certain criteria first, such as whether they face an “excessive burden” in receiving care at a Department of Veterans Affairs medical center. Under the current Choice guidelines, veterans also must wait at least 30 days for an appointment at a VA facility before they are eligible to receive care from a private doctor.

TRUMP: “And I used to say before I really was well-versed on the veteran situation in health care, I used to say all the time, ‘Why don’t they just let the folks go to a doctor?’ They’d wait in line for 7 days, 9 days, 14 days, 21 days … I said, ‘Why don’t they just let them go see a local private doctor, pay the bill and take care of it?’ And it’s turned out to be something that the veterans love. And it’s passed. It’s passed.” — remarks Tuesday.

THE FACTS: It’s not clear if veterans love the current Choice program, judging by the wait times. Despite the Choice program’s guarantee of providing appointments within 30 days, a recent report from the Government Accountability Office found that veterans actually waited an average of 51 to 64 days to receive care. A newly expanded Choice program will take at least a year to be implemented. ___

RUSSIA INVESTIGATION TRUMP: “Collusion is not a crime, but that doesn’t matter because there was No Collusion (except by Crooked Hillary and the Democrats)!” — tweet Tuesday.

TRUMP LAWYER RUDY GIULIANI: “I have been sitting here looking in the federal code trying to find collusion as a crime. … Collusion is not a crime.” — remarks Monday on Fox News.

THE FACTS: It is correct to say election collusion isn’t a precise legal term. The U.S. code mostly uses the term “collusion” in antitrust laws to address crimes like price fixing. As it relates to Russia and U.S. elections, the term can be seen as shorthand for plenty of violations of specific laws on the books.

For instance, there could be legal violations if Trump’s presidential campaign is found to have collaborated with Moscow, including a conspiracy to defraud the United States. There are also laws against election fraud, computer hacking, wire fraud and falsifying records, if those apply.

So far, special counsel Robert Mueller has accused the Russians of hacking into Democrats’ computers and stealing emails, as well as trying to stoke U.S. tensions before the 2016 election using social media.

Mueller might decide, for example, that a crime was committed if he finds evidence that an American was involved in the hack of Democrats, either by soliciting it or paying someone to do it. As well, a conspiracy to defraud the United States can be used to refer to any two people using “deceit, craft, or trickery” to interfere with governmental functions, such as an election.